The main difficulty of solving (REF ) arises from the atomic norm \(\Vert \mathbf {h} \Vert _\mathcal {A}\)  over the continuous Fourier dictionary \(\mathcal {A}\) .
Fortunately, the minimization of \(\Vert \mathbf {h} \Vert _\mathcal {A}\)  admits the following semidefinite program (SDP) thanks to the Carathéodory–Fejér–Pisarenko decomposition [1]}, [2]}
\(\arg \min _{\begin{array}{c}\mathbf {u} \in \mathbb {C}^N \\ v \in \mathbb {R}\end{array}}\frac{1}{2}\left(\frac{1}{N} \mathrm {tr}\big (T(\mathbf {u})\big ) + v \right)\hspace{10.0pt} \mathrm {s.t.} \hspace{10.0pt}\begin{bmatrix}T(\mathbf {u}) & \mathbf {h} \\\mathbf {h}^\mathrm {H} & v\end{bmatrix}\succeq 0,\) 
